Question

Factor the perfect squares. $$ x^{2}-10 x+25 $$

Short answer

(x - 5)^2

Step-by-step solution

Identify the form of a perfect square trinomial

Recognize that the expression is a trinomial in the form of \[ a^2 - 2ab + b^2 \].

Rewrite the expression in the form of a perfect square trinomial

Notice that the given expression can be written as \[ x^2 - 2 \times 5 \times x + 5^2 \]. Here, \( a = x \) and \( b = 5 \).

Write the trinomial as a squared binomial

Express the trinomial \[ x^2 - 10x + 25 \] as a product of \( (a - b)^2 \), which gives: \[ (x - 5)^2 \].

Key concepts

trinomial

A trinomial is a type of polynomial that consists of three terms. In the context of factoring perfect squares, identifying the three terms correctly is crucial.
For example, in the given exercise, we have:

• First term: The first term is x^2. This represents a^2 in the perfect square trinomial formula.
• Second term: The second term is -10x. This corresponds to -2ab.
• Third term:
  The third term is 25. This is equivalent to b^2.

Understanding these components helps in identifying and factoring perfect square trinomials efficiently.

Notice that a trinomial is characterized by three terms connected either by addition or subtraction, which is important for further steps in factoring.

perfect square

A perfect square trinomial is a special type of trinomial that can be expressed as the square of a binomial. This is important because it makes factoring more straightforward once you recognize the pattern.

The general form of a perfect square trinomial is:

• a^2 + 2ab + b^2 or
• a^2 - 2ab + b^2.

These can be factored into (a + b)^2 and (a - b)^2 respectively.

In our exercise's specific case, the trinomial x^2 - 10x + 25 fits into the (a - b)^2 pattern. Here, a = x and b = 5, making the factored form (x - 5)^2.

Recognizing the structure of perfect squares is key in simplifying complex expressions and solving various algebraic problems.

binomial

Binomials are polynomials containing exactly two terms. They play a pivotal role when dealing with perfect square trinomials because a trinomial can often be factored into the square of a binomial.

The structure of binomials used in perfect square trinomials typically looks like:

• Form: They can appear as (a + b) or (a - b).

In our exercise, recognizing that x^2 - 10x + 25
can be written as (x - 5)^2 is essential.

This works because once the trinomial is identified as a perfect square, it simplifies to the square of the corresponding binomial.

Understanding how to factor into binomials is an important skill in algebra, paving the way for more advanced topics and problem-solving methods.